Warning: This question will be fairly ill-posed. I have spent a lot of time trying to make it better posed without success, so please bear with me.
A single $SU(2)$ spin may be represented by the $0+1$ dimensional non-linear $\sigma$ model with target space $S^2$. This $\sigma$ model admits a Wess-Zumino-Witten term. Now WZW terms are very funny things. One way to think about is purely from the path integral side. The existence of the topological term comes from the topological structure of the path integral, and the quantization of the WZW term comes from the single valued-ness of the integral.
Another way to obtain the WZW term is to start from the Hilbert space of an $SU(2)$ spin (the "algebraic" side) and then construct the path integral. The fact that the WZW term is quantized and topological is totally natural from this perspective, since the WZW term has nothing to do with dynamics. It simply keeps track of which $SU(2)$ representation we have - so of course it doesn't renormalize or care about any details.
Further, instead of a single spin, we can consider a $d$ dimensional lattice spins. Under some conditions we can again write this as $\sigma$ model. There may be a WZW term depending on $d$. So there is a classification of possible WZW terms depending on dimension.
Here is the issue: The classification of WZW terms on the path integral side is very clear. It is just some appropriate dimensional homotopy group. What is being classified on the algebraic side? We are somehow counting distinct representations of the operator algebra - but what precisely?
Hopefully, that makes non-zero sense. What follows is a dump of my brain contents.
1) The question has nothing to do with symmetry. I will still have a topological term if add a nonzero Hamiltonian that breaks every symmetry. It is true that at long distances I may end up in different topological sector but this seems irrelevant.
2) A quantum system consists of two parts: an operator algebra (with a Hamiltonian) and a Hilbert space, which is a choice representation of this algebra - the WZW terms keeps track of this choice.
3) It is obviously sensitive to more information than the dimension of the Hilbert space since there are representations of $SU(2)$ and $SU(3)$ with the same dimension but the homotopy sequence of $S^2$ is different from that of $S^3$.
4) As far as I can tell there is nothing special about WZW terms. Any topological term should have some similar question, at least via the bulk-boundary correspondance. Since we can set $H=0$ the question is somehow about the algebraic analogue of Topological QFT. So someone who understands TQFT should be able to explain.
The closest I have been able to come to an answer is something like the following. To construct a coherent state path integral representation my Hamiltonian can only involve a certain algebra of operators. (I think.) For example, if I have a lattice with a four dimensional Hilbert space on each site, I can pretend that that is a lattice of spin $3/2$. But if I just have random matrix elements coupling neighboring sites then the coherent state path integral doesn't seem to come out well. It's only when the only couplings that appear in the Hamiltonian are the spin operators that I can perform the usual path integral manipulation.
So as I basically said before, the defining question is "what is the algebra of operators defined on a site that may appear in the Hamiltonian"? Again, as I said before there is no notion of symmetry in this definition, since the Hamiltonian does not have to lie in the center of this operator algebra. And again the spectrum of topological invariants is sensitive to the representations of this algebra.
Now to construct a path integral representation I would like to have coherent states. In the case that my algebra is just a finite Lie Algebra then this probably can be done using a root decomposition, and basically following the $SU(2)$ construction. So I get some geometric space, that I can probably read off the Dynkin diagram somehow. Then maybe by going backwards from the homotopy calculation I can figure whats happening on the algebraic side. So I guess its just the ADE classification of symmetric spaces, maybe?
In the case that my operator algebra is not a finite Lie algebra, I don't know, mostly because I know nothing about Algebra.
This post imported from StackExchange Physics at 2014-04-05 02:58 (UCT), posted by SE-user BebopButUnsteady